My first year in Madrid I made 65 euros/week teaching private lessons, and this year I made 63 euros/week when I taught all four classes a week. This extra money really helps out, and if you're working as an auxiliar there is definitely time during the week for evening private lessons. Below are some different ways to find private English lessons in Madrid.
Tus clases particularesThere is a website you can use (Tus clases particulares), which is how I got three of my four lessons this year. The next couple days after putting up my ad, I kept getting messages from interested people and in the end had to turn down lots of lessons and remove the ad after I had picked the first family. It's nice because you can sift through the messages of interested people and be a bit picky; I decided I only wanted classes within walking distance of my apartment so as not to spend lots of time in transit, and I found them.
Lingo bongoAnother website to occasionally check out is Lingo Bongo. Along with "private classes," there is also a section of "jobs offered," which includes language academies in the city. If in the spring you're looking for a summer job, this would be one place to look.
Facebook groupsIf you are working as an auxiliar de conversación, I would recommend joining the facebook group for Auxiliares in Madrid. I was in the group for this past year, and people would frequently post private classes that they couldn't take. So that can be another way to get private classes.
FlyersAnother option is putting up flyers. If there's a local library wherever you end up living, I'd for sure put one there. I put up flyers in the fall as well, but ultimately got my students from tusclasesparticulares. With the flyers too, you can keep it in the neighborhood where you want to give classes. If anyone wants a copy of the flyer I used, just send me an email (rebewithaclause [at] gmail [dot] com).
Also, keep your eye out for flyers posted by people looking for an English teacher. My first year in Madrid I had a friend who found a play-class with two younger children for 25/hr from a flyer!
CoworkersFinally, if you're an auxiliar, wherever you're working might also be an option to find private lessons. Let the teachers at your school know you teach private lessons, and it's quite possible that a teacher (of any subject) or students at your school want lessons.
Once you have one class, they seem to accumulate from there. One day I was in the elevator of a student's apartment building and the woman with me in the elevator started asking if I taught private English classes, and told me about her daughter who wanted classes. Also, the mother of one student all year kept asking if I had space for another student, because once her friends found out that her son had a native English teacher, they wanted classes too. I had to turn down all of those classes because I didn't have any extra time in the week, but just so you know once you get a good reputation with some students, it's very possible that more classes will come your way via word of mouth.
One last note: If you've never taught English in Europe before, be aware that you'll most likely be teaching British English!
Wondering how much you should charge? Want some tips to getting paid smoothly and keeping track of multiple students? I tackle those questions, among others, in the Teaching Private English Classes in Spain FAQ post below (click on the image):